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In level 52 we get a bunch of words that end with a consonant with tanween, followed by an alif. As far as I can tell, these words sound exactly as they would if the alif weren't there. What's the role of this alif at the end? Why is it there, and does it affect the pronunciation in any way?

Yes it affects the pronunciation when you pause or stop after the word (taking some breath) so the tanween goes and the alif is pronounced in that case. Unlike the other cases when there is a dhamma or kasra, where the pausa makes you drop the tanween and it becomes sukun.
There are also other pausa rules that affect only the pronunciation, like the "ة" pronounced "t" but when you pause, it becomes "h"

You are right. I felt that I didn't explain well.
In Arabic, the letters are consonants. Arabic uses vowel marks wich are not considered as letters. The vowel marks are dhamma (u), kasra (i) fat-hha (a/e) and sukun (no vowel). When a consonant is followed by a vowel, we put on it the corresponding vowel mark but when it is not followed by a vowel, we put the sukun mark instead. But in most of the writing, these marks are habitually omitted for the ease and speed of writing. We use them when we need some disambiguation. There is also the shadda mark wich indicates a doubled consonant.

Caution: In many styles of writing, when a letter has a shadda and kasra, the kasra is underneath the shadda instead of the whole letter. Be careful because you may mistake it with a fat-hha (it is on the top of the letter). But in handwriting, we often put the kasra under the whole letter
even with a shadda on top. When a letter has a shadda and a fat-hha, don't forget to put the fat-hha on the top of the shadda or it would be a kasra. Maybe this is because the shadda symbolizes the second letter of the two doubled consonants. Like the word "أُمِّي" (my mother), it reads ummii; this is a doubled consonant; instead of writing two mims, the first with a sukun and the second with a kasra, we write one mim with a shadda and a kasra. The shadda symbolizing the second mim, should take the kasra instead of the first mim. Try to paste the word "أُمِّي" in windows notepad, and you will see the kasra under the whole letter, while here it shows it under the shadda and on the top of the letter.

1-You should omit the final vowel for every breath of talking. Because we should not breathe just after a vowel, we should continue talking in the same breath until we want to breath (even a little) where the last vowel must be omitted in pronunciation.

Caution: This does not apply to long vowels because they are reinforced by a letter (alif, waaw or yaa') like if we write "kuw" instead of "ku" or "kiy" instead of "ki" so it ends in a consonant, that letter cannot be dropped. We take an example of a definite noun in the nominative case: al-ki-taa-bu (the book), it becomes al-ki-taab when you stop just after it.

2-The taa' marbuuta "ة" is pronunced like ت but when you stop just after it, it becomes pronounced like ه and its vowel is omitted. This letter is always final. It comes at the end of some words, mainly feminine nouns. It comes also at some kinds of plural, and even some masculine nouns or proper nouns. We can note also that this letter is always preceded by either:

(A) an alif or

(B) a letter with fat-hha.

Usually they break these rules in informal speach and use informal rules: In the case (A) they pronounce this letter as a normal ت even before a pausa! For example, the word توراة (Torah) where they read it as tawrat instead of tawrah. And the word حياة (life) where they read it as hhayat even before a pausa. In the case (B) they pronounce this letter as an alif extending the fat-hha before a pausa or omit it when they continue speaking! In the case where the word is "mudhaf ilayh" i.e. in a form like "end of the story" which is "نهاية الحكاية", they pronounce "nihayat lhhikaya" instead of "nihayatu lhhikayah". So they don't omit the letter if the word is "mudhaf ilayh". Please note that the formal rules are simpler to learn.

3-The tanween with kasra or dhamma also is omitted in the pausa. In other words, the unwritten nuun that we pronounce and that is symbolized by two fat-hhas, two dhammas or two kasras is omitted in the last word of every breath. We take some examples: ki-taa-bun becomes ki-taab; qa-la-mun becomes qa-lam; dar-sin becomes dars; wa-ra-qa-tin becomes wa-ra-qah

4-The tanween with fat-hha is always written with a final alif except when the word ends with "ة". When we continue speaking, the final alif is not pronounced and we pronounce the tanween normally. But when we stop speaking, the tanween is omitted and the alif is pronounced extending the fat-hha. When we stop after ki-taa-ban, it becomes ki-taa-baa but we can never say ki-taab. The words ending with "ة" act like when there is dhamma or kasra and do not have an alif whith the tanween of fat-hha. Like wa-ra-qa-tan which becomes wa-ra-qah when we stop after it and we cannot say wa-ra-qa-taa.

Caution: There are some words that do not have a written alif in the tanween of fat-hha but the alif is pronounced! unlike words who have a "ة" at the end. These words are the ones that end with an alif followed by a hamza. We cannot write a hamza between two alifs. But the second alif which is omitted in spelling, still be pronounced in pausa. We take an example: the word sa-maa-'an ("sky" in the accusative case) becomes sa-maa-'aa when you stop just after it but it is written "سَمَاءً", not "سَمَاءًا". Maybe this rule is because it is ugly when we write a hamza between two alifs. Often, these rules are not applied correctly by people.

If you want more advanced topics, these are two rules:"الْعَرَبُ لَا تَبْدَأُ بِسَاكِنٍ وَلَا تَقِفُ عَلَى مُتَحَرِّكٍ" which means that the Arabs don't begin a word with a letter with a sukun and don't end speech with a vowel. (we have discussed the second one above in pausa rules) For example, Britain in Arabic is "بَرِيطَانْيَا" (ba-ri-tan-ya) but not bri-tan-ya. This is because of the first rule. But sometimes in speaking, they may say bri-tan-ya according to the original pronounciation.

There are some words that have only one letter (consonant), like "عِ" pronounced 'i which is the verb to be aware/to realize/to perceive in the imperative. When it is the last word or single word, you pronounce it 'ih "عِهْ" and this haa' is called "هاء السكت" and is intended to protect the vowel from falling and leaving the consonant alone and unpronounceable. Even when the word is preceded by another word it still be added because a final word with a single letter can never occur in Arabic. This causes two rules to contradict: When a word with a single letter is at the end of speech, the first rule says that this letter being the first in a word, cannot have a sukun and the second rule says that this letter being in the end of speech cannot have a vowel. So will it have a vowel or a sukun?! That's why the haa' is used. There are other uses of this haa'. For example "why" in Arabic is "لِمَاذَا" or "لِمَ". If we would like to say "Why you did that?" we say: "لِمَ فَعَلْتَ هَذَا" but when we want to say only Why?", we say then: "لِمَهْ" (li-mah) because this word cannot be pronounced as lim.

There are also some confusing fonts where a miim which comes after a laam becomes a little thing hidden under the laam. In arabic fonts, not only the letters change shape but they interact with each other. Please note the difference between the two writings of the word "المبنى" (the building). Note the second miim shape under the laam and the nuun merged. We can see also how the yaa' is attached to the jiim in the word "يجمع" (collects).

Wow. You've explained a lot more than just the thing I asked about, and most of it is clear. I'm going to have to come back to this comment and re-read it every few weeks until it all sinks in.

You could make a new course focusing on Arabic spelling and pausa rules, introducing these concepts one by one and teaching a bunch of examples that illustrate each one, before building the next concept on and adding new examples that show it. I would love to take that course.

One question I have after my first reading of these comments: How do we know which word ends at the end of a breath, and which word we "continue speaking" after? Is it at the end of each sentence? Or after a certain number of syllables? Or after particular sounds or sound combinations? Or something people can do whenever they want, so that the same sentence may be said with different sounds because people may choose to say it with a pausa in different places?

Yes, it is indeed whenever you want to stop. It is like a full stop or a comma, but comes naturally, you don't need to force the pausa in any place. Maybe I will try to make a course about Arabic in the future. I don't know from where to start. I need to clarify many misused things. Mentioning everything is not enough, I should be clear. Also I need to be organized rather than mentioning miscellaneous things, therefore, I would be very slow. I shall be careful and try to minimize errors. I think, processing slowly and correctly is much better than processing quickly and incorrectly. Because a detailed explanatory course may only slow down the learning process while a course with incorrect or unclear things may cause one to get it wrong and to be confused and unsure and that would discourage people to continue such an unpleasant course until the end.